Candidates for non-sofic groups

What are the "simplest" examples of countable groups that are not known to be sofic?

• I think Thompson's groups are not known to be sofic. See Pestov's survey: google.com/… – Ian Agol Feb 16 '14 at 5:56

$\langle a,b,c,d\mid a^b=a^2, b^c=b^2, c^d=c^2, d^a=d^2\rangle$
(where, as usual, $a^b$ means $b^{-1}ab$). Terry Tao wrote a nice blog post about it here.
• Soficity is preserved under direct products, so just take a direct product of non-amenable and non residually finite sofic groups, such as a free group and a Baumslag-Solitar group like $\langle x,y \mid y^{-1}x^2y=x^3 \rangle$. – Derek Holt Feb 10 '14 at 18:21